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On Courant's nodal domain theorem
We study the validity of Courant's nodal domain theorem for eigenfunctions of selfadjoint second order elliptic operators with low regularity assumptions on the coefficients. We prove that in two dimensions Courant's theorem holds also when the coefficients are just bounded and measurable. In the higher dimensional case, we prove a weakened version of Courant's theorem when the coefficients in the principal part are Hölder continuous.
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